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ON  THE  PENTACARDIOID 


ABSTRACT  OF  A 

DISSERTATION 


Submitted  to  the  Board  of  University  Studies  of  the  Johns  Hopkins 

University  in  conformity  with  the  requirements  for  the 

degree  of  Doctor  of  Philosophy 


BY 

EDWARD  C.  PHILLIPS,  S.  J. 

March,  1908 


BALTIMORE,  MD.,  U.  S.  A. 
1909 


ON  THE  PENTACARDIOID 


ABSTRACT  OF  A 

DISSERTATION 


Submitted  to  the  Board  of  University  Studies  of  the   Johns  Hopkins 

University  in  conformity  with  the  requirements  for  the 

degree  of  Doctor  of  Philosophy 


BY 

EDWARD  C.  PHILLIPS,  S.  J. 
March,  1908 


-■      OF    1  HE 

UNIVERSITI. 

OF 
^ALIFOf 


BALTIMORE,  MD.,  U.  S.  A. 
1909 


/ 


The  writer  desires  to  express  his  gratitude  for  the 
constant  interest  and  kindly  encouragement  and -assist- 
ance given  to  him  by  Professor  Morley,  not  only  "during 
the  preparation  of  this  paper  but  during  his  entire 
course  at  the  University. 


ON  THE  PENTACAliDIOID. 

Bt  Edward  C.  Phillips.  S.  J. 


In  the  discussion  of  the  metrical  properties  of  finite  systems  of  lines  in  a 
plane  there  arises  a  series  of  curves  called  by  Professor  Morley  Ennacardioids;* 
these  curves,  over  and  above  their  usefulness  in  connection  with  the  system  of 
lines  from  which  they  arise,  have  many  interesting  properties  which  seem 
worthy  of  some  special  investigation  ;  and  it  is  proposed  in  this  article  to  make  a 
study  of  the  Pentacardioid,  the  first  in  the  series  of  Ennacardioids  which  has 
not  as  yet  received  any  detailed  treatment.  The  general  symbol  for  an  Bnna- 
cardioid  is  C",  and  we  shall  use  throughout  this  paper  the  corresponding  symbol 
C^  for  the  Pehtacardioid. 

The  system  of  coordinates  and  the  general  method  of  analysis  which  will  be 
employed  are  those  described  in  §  1  of  Professor  Morley's  Memoir  on  Reflexive 
Geometry.  However,  as  the  notation  in  the  various  articles  on  this  and  allied 
subjects  varies  considerably,  I  shall,  in  order  to  avoid  confusion,  here  state 
briefly  the  symbolism  I  intend  to  follow.  The  conjugate  of  a  complex  number  x 
will  be  represented  by  the  letter  y,  and  the  conjugate  of  a,  by  h.  A  complex 
number  of  absolute  value  equal  to  unity  is  called  a  turn  and  will  in  general  be 
denoted  by  the  letter  <  or  by  a  Greek  letter;  the  conjugate  of  a  turn  being  its 
reciprocal  needs  no  special  notation.  There  are  certain  special  turns  to  which 
definite  symbols  have  already  been  assigned,  and  these  I  shall  retain  ;  they  are 
the  following  :  The  square  root  of  negative  unity,  designated  by  i  ;  the  cube  roots 
of  unity,  designated  by  w  and  its  powers  ;  and  the  fifth  roots  of  unity,  designated 
by  £  and  its  powers  Thus  these  three  symbols  are  defined  by  the  equations 
i^  +  1  =  0,  o'  —  1=0,  and  e^  —  1  =  0.  The  modulus,  or  absolute  value,  of 
a  complex  number  will  be  denoted  by  the  letter  p,  or  by  placing  the  number 
between  parallel  strokes,  thus  :   |  a;  | .   A  turn  may  also  be  expressed  exponentially, 

*F.  Morley:   On  Reflexive  Geometry,  Transactiont  of  the  American  Mathematical  Society,  Vol.  VIII  (1907), 
pp.  15  ff. 


186967 


2  ON  THE  PENTACARDIOID. 

and  thus  any  complex  number  is  of  the  form  ft,  or  pe**,  where  6  is  the  amplitude 
of  the  complex  number. 

An  equation  in  complex  variables  is  said  to  be  self-conjugate  when  it  is 
identical  with  the  equation  obtained  by  replacing  each  quantity  by  its  conjugate, 
or  differs  from  this  equation  merely  by  some  factor;  such  an  equation  corre- 
sponds to  an  equation  in  real  variables  with  real  coefficients.  The  roots  of  a 
self-conjugate  equation  in  t  are  either  turns  or  pairs  of  inverse  points  as  to  the 
base  circle,  and  these  latter  bear  to  the  self-conjugate  equation  in  i  the  same 
relation  that  pairs  of  conjugate  imaginary  roots  bear  to  the  equation  with  real 
coefficients  in  one  real  variable. 

§  1.      The  Equation  of  the  Curve. 

The  Pentacardioid,  or  C"*,  as  defined  by  its  line  equation  in  its  most  general 
form,  is 

X—  ao  -f  ba^t  -f  IQa^^  +  \Qh.f  +  bh^t^  +  {V  —  hY  =  0.  (1) 


'» 


The  turn  t  is  here  used  as  a  parameter ;  each  value  of  t  gives,  for  a  varying  x,  a 

line  or  tangent  of  the  curve,  and  the  envelope  of  this  system  of  lines  is  the  point 

curve.     The   equation   of  the  point  curve  can  be  obtained    in    map   form    by 

eliminating  «/ between  equation  (1)  and  its  derivative  as  to  t\  the  elimination 

gives 

x  =  ao— 4ai<— 6a2<2— 462<»  — 6i<*.  (2) 

The  point  Oq  is  the  singular  focus  of  the  curve  and  will  be  called  its  center. 

There  are  in  the  above  equation  three  independent  complex  arbitFary  con- 
stants, and  hence  a  C'°  depends  on  six  conditions.  It  takes  two  complex  constants, 
equivalent  to  four  conditions,  to  fix  the  reference  system  so  that  there  are  only 
two  absolute  conditions  required  to  fix  the  shape  of  a  C^  I  shall  in  general 
consider  two  curves  to  be  the  same  when  they  are  similar ;  or,  stating  the  same 
thing  analytically,  two  curves  z=if(t)  and  x=.f^{t)  will  be  considered  the 
same  when  they  are  so  related  that  there  exists  between  the  points  of  the  two 
curves  a  one-to-one  correspondence  of  the  form  z  =  mx  -\-  n,  where  m  and  n  are 
arbitrary  complex  constants.  A  correspondence  or  transformation  of  this  kind 
may  be  called  a  proportion ;  it  leaves  the  shapes  of  figures  entirely  unaltered, 
merely  changing  their  size  and  position  in  the  plane. 

In  the  method  of  discussion  which  I  shall  follow,  it  will  be  found  convenient 
to  take  the  center  of  the  curve  as  the  origin  and  to  make  a^  equal  to  unity  ;  the 


ON  THE  PENTACAKDIOID.  3 

equation  in  this  form  will  be  called  the  standard  equation.  The  standard  line 
and  map  equations  of  the  general  (7^  are  therefore  : 

x+ 5t+  lOat^  +  10bt^+ ^t^ +  i/t^  =  0,  (3) 

x  =  —  it—6at^—'ibt^  —  t\  (4) 

The  equations  in  this  form  evidently  depend  upon  and  are  entirely  determined 
by  the  single  complex  number  a,  which  T  shall  therefore  call  the  determining 
coefficient  of  the  curve.  The  equation  of  any  C^  can  be  reduced  to  this  form* 
by  means  of  a  proportion ;  for  let  the  equation  of  the  curve  be  given  in  its  most 
general  form 

z  =  Oo  —  "^pi^it  —  Gp.^xr,t^  —  4(p2/x2)<^  —  {pi/xi)t^ ; 

if  in  this  we  replace  t  by  xfH  (a  change  which  leaves  both  the  curve  and  the 

reference  system  unaltered),  and  then  divide  by  p^xf^  and  transpose  the  constant 

term,  we  have : 

(z  —  ao)/pi4'^  =  —  4<  —  6pxt^  —  4{p/xy  —  t* 


fi 


where  p^pg/pi  and  x=:x^lx\^.     And  now  the  equation  is  in  the  standard  form. 

This  reduction  is  unique  excepting  as  to  the  ambiguity  introduced  by  the 
process  of  taking  the  cube  root  of  xj,  and  hence  it  can  be  made  in  three,  and  only 
three,  ways.    We  thus  arrive  at  the  following  important  theorem : 

Two  G^'s  are  similar  when,  and  only  when,  the  determining  coefficient  in  the 
standard  equation  of  one  curve  differs  from  the  determining  coefficient  of  the  other 
curve  at  most  by  a  cube  root  of  unity. 

Furthermore,  since  replacing  a  number  (or  a  point  which  the  number  repre- 
sents) by  its  conjugate  is  equivalent  to  a  reflexion  in  the  axis  of  reals,  it  follows 
that  if  the  determining  coefficient  of  one  G^  differs  from  the  conjugate  of  the  deter- 
mining coefficient  of  another  C"  at  most  by  a  cube  root  of  unity,  the  two  curves  are 
inversely  similar  ;  i.  e.,  they  are  reflexions  of  each  other  in  the  axis  of  reals  and 
so  bear  to  each  other  the  relation  of  an  object  and  its  image  in  a  plane  mirror. 

§  2.      Construction  of  the  Curve. 

A  construction  for  any  C"  depending  on  certain  properties  of  its  osculants 
was  described  by  Professor  Morley  in  §  6  of  the  Memoir  on  Reflexive  Geometry 

•There  1b  the  one  exceptional  case  of  the  cnrves  for  which  the  coeflBcient  Oj  In  equation  (2)  la  zero,  but 
we  can  consider  this  as  a  limiting  case  and  include  it  In  our  standard  equation  by  allowing  the  determiniDg 
coefficient  a  to  take  the  special  value  co  ;  and  It  should  be  borne  in  mind  that  \a\  is  really  the  ratio 
of   |a,|  to  \a^\. 


4  ON  THE  PENTACARDIOID. 

already  referred  to  ;  but  there  is  another  method,  also  mentioned  by  Professor 
Morley,*  which  is  more  easily  applied  to  the  case  of  the  C^  and  I  shall  here  set 
forth  a  development  of  this  second  method.  Refer  to  the  standard  equation  of 
the  C^  and  consider  the  two  special  curves  corresponding  respectively  to  the 
values  of  the  determining  coefficient  a  =  0  and  a-=-  (*:> .     They  are 

Xi  +  5<+5<*  +  j/i<«=0,  (5) 

a;3  +  10<'=+ 10<^  +  y,<«  =  0;  (6) 

with  map  equations 

x,  =  —  U-t\  (7) 

x^^  —  Qe—U^.  (8) 

These  curves  are  two  epicycloids  easily  constructed  by  simple  mechanical 
means ;  the  first  is  the  three-cusped  epicycloid  traced  out  by  a  point  on  the  cir- 
cumference of  a  unit  circle  rolling  about  a  circle  of  radius  3,  whilst  the  second  is 
the  one-cusped  epicycloid  traced  out  by  a  point  on  the  circumference  of  a  circle 
of  radius  4  rolling  about  a  circle  of  radius  2.  From  these  two  curves  all  other 
C^'s  can  be  built  up  very  simply.  We  must  first  notice  that  any  curve  of  the 
form 

x.i=  —  Qat^—Aht^  (9) 

is  derived  from  (8)  by  multiplying  x^  by  px^,  where  a  =  pz  =  pe".    For  if  in  (9)  we 
replace  t  by  xH,  we  have 


aja 


—  QpxH^  —  4px7'  =  fx% .  (10) 

Now  the  effect  of  the  factor  px^  on  the  curve  is  merely  to  rotate  it  through  the 
angle  50  and  to  enlarge  it  in  the  ratio  p  :  1.  Looking  again  at  the  standard 
equation  (4)  of  the  general  C^,  we  see  that  it  is  the  sum  of  two  simpler  equations, 
namely  (7)  and  (9) ;  therefore  any  C^  can  be  constructed  by  adding  the  vectors 
that  join  the  origin  with  corresponding  points  of  two  epicycloids  properly  placed. 
By  corresponding  points  are  meant  points  on  the  two  curves  given  by  the  same 
value  of  t.  Since  the  clinant  of  the  tangent  to  all  these  curves  at  the  point 
given  by  the  parameter  t  is  — t^,  it  is  evident  that  at  corresponding  points  of 
the  two  curves  (7)  and  (9)  the  tangents  are  parallel,  and  owing  to  this  fact  we 
can  readily  determine  as  many  pairs  of  corresponding  points  as  we  wish.  We 
must,  however,  choose  the  proper  starting  points  on  the  two  curves,  for  there 


♦Orthocentrlc  Properties  of  the  Plane  »4-Llne :    Transactions  of  the  American  Mathematical  Society,  Vol.  IV 
(1903),  pp.  7  and  8. 


a 


a 


^ 


(6) 


6  ON  THE  PENTACAEDIOID. 

are  five  tangents  to  a  G^  in  any  given  direction,  and  according  as  we  associate 
one  or  another  of  these  five  points  on  the  second  curve  with  a  selected  point  on 
the  first  we  obtain  five  different  resulting  C^'s.  This  ambiguity  can  be  obviated 
by  choosing  a  definite  value  of  the  parameter  t  and  determining  the  points  given 
by  this  value  on  the  two  curves  by  the  ordinary  process  of  plotting.  Thus,  putting 
t  =  1,  say,  we  have  aij  =  —  3  and  a-g  =  —  6a  —  46  ;  these  points  are  very  easily 
found  as  soon  as  we  know  the  value  of  a,  and  taking  this  pair  as  our  starting 
points  all  further  ambiguity  is  impossible. 

In  practice,  it  is  much  more  convenient  to  take  the  mean  of  two  points  than 
to  take  their  sum,  since  to  get  the  mean  we  need  merely  bisect  the  segment 
joining  the  points;  and  for  our  purpose  the  points  (xj  -\-  x^l2  are  just  as  good  as 
the  points  (xj  +  a-g).  Moreover,  if  we  wish  to  alter  the  relative  sizes  of  the  two 
fundamental  curves,  which  is  equivalent  to  altering  the  modulus  of  the  deter- 
mining coeflicient  of  the  resulting  curve,  all  we  need  do  is  to  divide  the  stroke 
joining  corresponding  points  in  a  different  ratio  than  1:1;  thus  if  we  wish  the 
curve  resulting  from  compounding  (7)  with  the  curve  a;  =  —  6p<^ — 4p<^,  we 
divide  the  stroke  joining  corresponding  points  of  (7)  and  (8)  in  the  ratio  p:  1. 
Finally,  when  we  wish  to  alter  the  amplitude  of  a,  we  merely  rotate  one  of  the 
curves  (7)  or  (9)  through  an  angle  equal  to  five  times  the  desired  change  in  the 
amplitude.  The  combination  of  these  two  changes  gives  us  every  possible 
variation  in  the  value  of  the  determining  coefficient,  and  this  method  has  been 
found  in  practice  to  be  a  very  simple  and  convenient  one  for  obtaining  all  the 
distinctive  types  of  C^  The  shifting  of  one  or  both  the  fundamental  curves 
without  rotation  does  not  affect  the  shape  or  size  of  the  resulting  curve  and  it  is 
more  convenient  to  place  the  component  curves  some  distance  apart  than  to  have 
them  concentric. 

Figure  II  shows  a  number  of  C^'s  constructed  by  this  method,*  and  Figure  I 
illustrates  clearly  the  carrying  out  of  the  method  in  two  particular  cases.  The 
two  broken  curves  (a)  and  {d)  are  the  fundamental  curves  in  this  particular  case ; 
their  equations  are  (disregarding  the  constant  term  which  has  no  effect  on  the 
shape  or  effective  position) : 

xi  =  —  3^  —  2^ curve  (a), 

arg  =  +  4<  +  <*    curve  {d). 

*  The  curves  in  Figure  II  are  free-liaud  reproductions,  on  a  reduced  scale,  of  larger  figures  constructed  In 
the  same  manner  as  Figure  I. 


M       S.^^/g-'e. 


•fV 


(il      S,  =  U" 


(c)     S.'i'/v^c" 


U)      S,  =  31>,  it" 


(e)      S,  =   fiit'^ 


(/)       S.-.^/^c'e 


b)  s.^iU^ 


(c-J     S,  =  %U  c'l^ 


U)  s,  =  :^'ixL^c> 


-t"  c  g XX.Y e,  E. 


')    -S.  -    6ci£.- 


(7) 


8 


ON  THE  PBNTACAEDIOID. 


On  each  of  these  curves  are  marked  thirty  points  such  that  the  angle  between 

the  tangents  at  any  two  successive  points  is  30  degrees ;  the  tangents  at  the 

points  1,  7,  13,  19,  25  on  either  curve  are  all  parallel  to  the  'tangent  at  the 

point  1  on  the  other  curve,  and  hence  any  one  of  those  five  points  could  be  taken 

as  a  corresponding  point  to  the  point  1  of  the  other  curve ;  the  points  marked  1 

on  the  two  curves  were  chosen  as  a  corresponding  pair.     The  strokes  joining 

these  two  points  and  the  other  corresponding  points  were  divided  in  the  ratio 

1  : 2,   and  thus  were  obtained  thirty  points  of  the  curve   (6),    of  which   the 

equation  is 

3a;  =  2a;i  -\- x^  z=  4t  —  6f  —  'ifi  +  <*. 

Next  the  strokes  were  divided  in  the  ratio  2:1,  giving  the  resulting  curve  (c), 
of  which  the  equation  is 

Sx  =  Xi  +  2x2  =  St—Sf—  2f+  2tK 

The  values  of  the  determining  coefficient  for  the  four  curves,  taken  in  the  order 
(a),  (J),  (c),  {d),  are  respectively  oo ,  — 1,  — 1/4,  0. 

§  3.      The  Singularities  of  the  C^. 

In  order  to  determine  completely  the  singularities,  we  need  the  ordinary 
equation  of  the  curve,  namely,  an  algebraic  relation  between  x  and  y  independent 
of  <;  we  can  best  secure  this  by  eliminating  t  between  equation  (2)  and  its  con- 
jugate. Performing  the  elimination  by  Sylvester's  dialytic  method,  we  arrive 
at  the  desired  equation  in  the  following  determinant  form: 

1 

y 

4b 

4 

6a 

6b 

4 

4a 

Since  x  and  y  appear  only  in  alternate  rows,  and  one  in  each  column,  the 
developed  equation  will  have  as  its  highest  term  a;*?/*;  hence  a  C"  is  of  order  eight. 
Since  the  four  highest  powers  of  x,  and  also  of  i/,  are  absent,  the  curve  will  have 
a  four-fold  point  at  infinity  on  each  of  the  axes  a;  =  0  and  y^=0;  i.  e.,  at  the 


X 

4 

6a 

46 

1 

4a 

6b 

4 

0 

X 

4 

6a 

0 

1 

•4a 

66 

0 

0 

X 

4 

0 

0 

1 

4a 

0 

0 

0 

X 

0 

0 

0 

1 

0 

0 

0 

0 

0 

0 

1 

0 

0 

y 

0 

0 

46 

1 

0 

4 

y 

0 

6a 

46 

1 

66 

4 

y 

=  0. 


(11) 


ON  THE  PENTACAEDIOID.  9 

■  circular  points,  /and  J.  The  symmetry  of  (11),  with  respect  to  x  and  y,  shows 
that  the  curve  is  symmetric  in  its  relations  to  the  two  circular  points.  A  four- 
fold point  being  equivalent  to  six  simple  double  points,  these  two  singularities 
are  equivalent  to  12  double  points;  and  as  a  rational  curve  of  order  eight  has  21 
double  points,  there  must  be  nine  other  double  points,  and  in  general  these  all 
lie  in  the  finite  part  of  the  plane.  Now  the  condition  for  a  cusp  is  the  vanishing 
of  the  derivative  of  x  as  to  <,  or : 

1  +  3a^  +  3i<2  + <3  =  o.  (12) 

As  this  is  a  self-conjugate  equation  in  t,  it  has  in  general  three  turns  as  its  roots; 
and  hence  every  C*  has  three  cusps,  and  only  three,  of  which  two,  however,  may 
be  imaginary.  Thus  the  nine  double  points  comprise  3  cusps  and  6  nodes. 
We  are  now  in  a  position  to  resolve  the  singularities  at  /and  /;  for  if  we  put 
d  =  number  of  nodes  at  each  of  the  circular  points,  and  Ic  =  number  of  cusps, 
and  note  that  the  C^  is  of  class  five,  as  is  shown  immediately  by  equation  (1), 
then  Plucker's  equation  connecting  the  order  and  class  of  a  curve  gives  us 

5  =  56  —  2{2d  +  6)  —  3{2k  +  3),  or  2d  +  3k  =  15.  (13) 

Moreover,  as  we  have  seen,  d  +  k  =  6,  and  hence  d  =  k  =  3;  so  that  each  of 
the  four-fold  points  is  equivalent  to  three  nodes  and  three  cusps. 

With  regard  to  the  line  singularities,  it  is  to  be  noticed  that  the  clinant, 
being  equal  to  — t^,  has  as  its  derivative  — 5t*,  and  as  this  can  not  vanish  for  a 
turn,  the  C^  has  no  inflections;  and  by  using  the  appropriate  Pliicker  equation, 
we  find  that  there  are  six  double  tangents.  This  completes  the  enumeration  of 
the  simple  singularities  of  the  curve. 

By  precisely  analogous  argument  we  can  show  that  the  general  C"  is  of 
class  n  and  order  2(n  —  1),  that  it  has  an  (w  — l)-fold  point  equivalent  to  n  —  2 
cusps  and  (n  — 2)(n  —  3)/2  nodes  at  each  of  the  circular  points,  that  it  has 
n  —  2  cusps  and  (n  —  2)(n — 3)  nodes  in  the  finite  part  of  the  plane,  and  finally 
that  it  has  (n  — 1)(«  —  2)/2  double  tangents  and  no  inflections. 

It  may  of  course  happen  that  some  of  these  singularities  besides  those  at  / 
and  J  are  imaginary ;  and  I  would  note  that  there  are  two  views  which  may  be 
taken  of  such  cases :  The  first  is  the  projective  view,  which  has  been  followed  in 
the  above  discussion  and  treats  all  the  singularities,  both  real  and  imaginary, 
as  on  the  same  footing;  the  second  view  is  the  one  more  proper  to  the  present 
method  of  analysis,  which  may  be  called  metrical  geometry,  and  which  acknowl- 
edges as  properly  belonging  to  the  curve  only  the  real  nou-isolated  singularities. 


10  ON  THE  PENTACAEDIOID. 

In  what  follows,  unless  the  contrary  is  stated,  I  shall  confine  myself  to  the  second 

view  and  deal  only  with  the  real  non-isolated  singularities,  and  one  of  my  main 

objects  will  be  to  determine  how  many  such  belong  to  any  given  curve  and  what 

effect  their  presence  may  have  on  the  form  or  shape  of  the  curve. 

Let  us  first  consider  the  cusps.    As  we  have  seen,  the  cusp  parameters  are 

given  by  the  self- conjugate  cubic  (12).     Since  this  is  of  odd  degree,  one  of  its 

roots  is  necessarily  a  turn  and  hence  every  G^  has  at  least  one  cusp.     The  other 

two  roots  of  (12)  may  be  either  turns  or  a  pair  of  inverse  points;  in  the  latter 

case,  the  G^  has  only  one  (real)  cusp.     If  the  three  roots  are  turns,  we  have 

three  cusps,  in  general  distinct;  two  roots  of  (12)  may  become  equal  for  special 

values  of  the  determining  coeflBcient  a,  and  in  this  case  two  cusps  come  together; 

the  resulting  singularity,  as  will  be  shown  immediately,  involves  also  a  node, 

and  is  a  triple  point  with  three  coincident  tangents  and  through  which  there 

passes  only  a  single  branch  of  the  curve.   Such  a  point  does  not  differ  materially 

in  appearance  from  an  ordinary  point  of  the  curve.     For  the  proof  it  will  be 

found  convenient  to  have  the  standard  equation  expressed  in  terms  of  the  cusp 

parameters.     From  equation  (12)   it   is   plain   that   if  t^,  t^,  ta  are   the  cusp 

parameters,  then 

ti  +  tz  +  t3  =  si  =  —  36, 

hh  +  ¥3  +  ¥1  =  Sg  =  3a,    ■  (14) 

Hence  the  standard  equation  may  be  expressed  in  the  form 

Sx  =  —  12t  —  6s/  -f  4Sjf  —  3<*.  (15) 

This  equation  shows  incidentally  that  the  shape  of  the  curve  is  determined  uniquely 
as  soon  as  the  cusp  parameters  are  given. 

If  two  roots  of  (12)  become  equal,  then  t^  =^2  ^^^  ^3  =  —  1/^!  so  that  the 
equation  of  the  curve  is,  in  this  case, 

Bx  =  —  -[2t  —  6{tl—  2lfiy  +  4{2ti—l/^y—3t\  (16) 

and  the  double  cusp,  given  hy  t  =  ti,  is 

3c  =  —  4ti—t\.  (17) 

A  pencil  of  lines  through  this  point  is  given  by  the  equation 

3c—3x  —  r{Sd—3y)  =  0,  (18) 

T  being  the  variable  clinant  and  d  the  conjugate  of  c.     If  we  substitute  in  this 
equation  the  values  of  a;  and  c  given  by  (16)  and  (17),  we  get  an  octavic  in  t  the 


ON  THE  PENTACAEDIOID.  11 

roots  of  which  are  the  parameters  of  the  eight  points  in  which  the  line  intersects 
the  C*;  the  result  is 

(<-0'(4/«?  +  ^  +  0-<iA-i/OW  +  iA+i/0  =  o,         (19) 

and  hence  t  =  ti  is  a  triple  root  for  all  values  of  r,  so  that  the  point  is  a  triple 
point.  In  like  manner,  by  substituting  the  values  of  s^,  Sg  and  x  in  the  line 
equation,  namely, 

3x+  15t^  lOss^—  10s/  +  15<*  +  Syt^=0,  (20) 

we  find  the  clinants  of  the  tangents  passing  through  this  point  to  be  given  by 

hence  the  three  tangents  at  the  triple  point  in  question  coincide. 

It  may  further  happen  that  the  three  roots  of  (12)  are  equal;  in  this  case 
the  three  cusps  unite  along  with  three  nodes  and  form  a  quadruple  point  with 
four  coincident  tangents.*  Such  a  point  does  not  differ  in  appearance  from  an 
ordinary  cusp. 

The  Nodes.  The  analytic  condition  for  a  node  on  a  curve  given  by  an 
equation  of  the  form  x=/{t)  is  evidently  [/(O — f{h)Vi^i  —  h)  —  ^-  In  the 
case  of  the  C^  we  have  then,  from  equation  (15), 

12  +  6*2(^1  +  0  —  45j(<f  +  t,t,  +  4)  +  3(<f  +  t%  +  titl  +  <i)  =  0.         (21) 

This,  along  with  its  conjugate,  which  is  of  degree  six,  namely 

12tltl—  QsMih  +  Q  +  WMti  +  tit,  +  4)  +  3(<f  +  4t2  +  tA  +  4)  =  0,     (22) 

if  solved  for  t^  and  ^ ,  would  give  us  the  eighteen  parameters  of  the  nine  double 
points;  but  this  direct  method  of  attack  involves  algebraical  difficulties  which 
preclude  any  reasonable  hope  of  success.  I  have  therefore  abandoned  any  direct 
investigation  of  the  nodes;  an  indirect  method  leading  to  a  partial  solution  of 
the  problem  will  be  described  further  on  (in  the  following  section). 

With  regard  to  the  combination  of  simple  double  points  into  higher  singu- 
larities, we  have  the  following  limitation :  Since  the  G^  is  of  class  five,  there 
can  not  be  any  proper  multiple  points  of  order  higher  than  two;  i.  e.,  no  points 

*The  proof  that  a  triple  root  of  (12)  does  produce  such  a  quadruple  point  on  the  C*  Is  entirely  analogous 
to  that  given  in  the  case  of  a  double  root,  and  therefore  need  scarcely  be  given  here. 


12  ON  THE  PENTACAEDIOID. 

through  which  pass  more  than  two  distinct  branches  of  the  curve.  Of  improper 
higher  multiple  points  we  have  three  kinds  : 

(a)  A  triple  point  through  which  there  passes  a  single  branch  of  the  curve ; 
this  results  from  the  combination  of  one  node  and  two  cusps ; 

(b)  A  triple  point  through  which  there  pass  two  branches  of  the  curve ; 
this  results  from  the  combination  of  two  nodes  and  one  cusp; 

(c)  A  quadruple  point  through  which  there  passes  a  single  branch  of  the 
curve. 

The  singularities  (a)  and  (b)  impose  but  one  condition  on  the  curve  and  hence 
there  is  a  single  infinity  of  C^'s  with  each  of  these  singularities ;  (c)  requires  that 
(12)  be  a  perfect  cube,  so  that  there  is  only  one  form  of  C^  which  has  a  quadruple 
point. 

It  can  happen  that  two  nodes  come  together  without  further  complications, 
in  which  case  we  have  a  tacnode.  This  case  leads  to  some  results  of  special 
interest.  Equation  (21)  is  the  general  condition  which  the  parameters  of  a  node 
must  satisfy ;  for  a  tacnode  they  must  satisfy  the  further  condition  <2  =  e% 
imposed  upon  them  by  the  fact  that  the  two  branches  of  the  curve  are  now 
parallel.  Combining  this  condition  with  (21),  we  have  the  necessary  and 
suflScient  condition  for  a  tacnode ;  namely, 

12  +  6s2(l+e")«— 4si(l+e"+e''")<2+3(l  +  e"+e^"+e^")<^=0,  72  =  1,2,3,4,  (23)„ 
or  in  somewhat  more  convenient  form, 

12+  6S2(  1  +  e^)t  +  4si£'"(  1  +  f")<2  —  3£*"f  =  0.  (24)„ 

Whenever  this  is  satisfied  by  some  value  of  t  which  is  a  turn,  the  C^  will  have 
a  tacnode;  as  the  equation  is  not  self-conjugate  except  for  special  values  of  Sj, 
we  must  determine  what  values  of  «i  will  cause  (24)„  to  have  a  self-conjugate 
factor.  There  are  several  methods  for  determining  this  condition  on  «i,  but  the 
best  is  to  eliminate  Sg  between  this  equation  and  its  conjugate  formed  on  the 
supposition  that  t  is  a  turn  and  then  to  solve  the  resulting  equation  for  Sj  in  terms 
of  t;  this  gives  us  the  map  equation  of  a  curve  such  that  when  s^  is  any  point 
on  it  the  corresponding  G^  will  have  a  tacnode.     The  value  of  s^  obtained  in  the 

above  way  is 

*,  =  3(1  If  +  2e''»«)/  2(f'"  +  6<").  (25)„ 

If  we  replace  t  by  e^"<  and  remember  that  l/(e^"  +  e'")  =  —  (e"  +  e*"),  this  can 
be  put  in  the  form 

Si  =  -(3/2)(6"  +  6*")(l/<^  +  20.    ■  (26)„ 


■      ON  THE  PBNTACARDIOID.  13 

Giving  n  its  four  values,  we  get  only  two  curves,  since  e"  +  e*"  takes  only  the 
two  distinct  values  e  +  e^  and  e^  +  e' »  each  of  these  two  curves  is  a  three-cusped 
hypocycloid,  or,  as  it  is  sometimes  called,  a  deltoid.  When  Sj  is  a  point  on  either 
of  these  curves,  given  say  by  the  value  t=zti,  then  (24)„  is  satisfied  identically 
by  the  value  t:=ti;  and  as  ti  is  a  turn,  it  follows  that  the  corresponding  C" 
has  a  tacnode. 

Finally,  three  nodes  can  combine  into  a  double  point  of  a  special  kind 
known  as  an  oscnode ;  at  such  a  point  the  two  branches  of  the  curve  have  three- 
point  contact,  whereas  at  a  tacnode  they  have  only  two-point  contact;  similarly 
the  tacnodal  tangent  is  equivalent  to  two  double  tangents,  the  point  of  contact 
counting  as  four  points  of  intersection  with  the  curve ;  whilst  the  oscnodal 
tangent  is  equivalent  to  three  double  tangents,  and  the  point  of  contact  counts 
as  six  intersections  with  the  curve.  There  are  two  G^'s  possessing  such  a 
singularity;  the  corresponding  values  of  Sj  will  be  derived  in  the  next  paragraph. 

The  Double  Tangents.  "We  take  the  line  equation  in  terms  of  the  cusp 
parameters ;  namely, 

3x+  15t+  10s/—  10s/  +15^*  +  Syt^  =  0.  (20) 

If  we  call  ti  and  ^g  the  parameters  of  the  points  of  contact  of  a  double  tangent, 
these  parameters  must  satisfy  the  two  following  relations : 

15^1  +  10s/—  10s/  +  15t\  =  15^2  +  10s/-  lOsi^  +  15<*,  ]  .^7) 

q  =  <2  >  J 

with  the  condition  ^i^^g.  Combining  these  two  relations,  we  get  the  four 
following  cubics  whose  roots  give  the  twelve  parameters  of  the  six  double 
tangents : 

3  +  2S2(1  +  £")<  +  2sj(e3»  +  e*")<2  —  Se'^'f  =  0 ;         n  =  1,  2,  3,  4.         (28)„ 

It  should  be  noted  that  the  three  roots  of  (28)^  are  equal  to  the  three  roots  of 
(28)j,  each  multiplied  by  s;  and  the  three  roots  of  (28)3  are  equal  to  those  of 
(28)2,  each  multiplied  by  e^  Thus  the  six  double  tangents  are  separated  into 
two  sets  of  three  which  are  distinguished  from  each  other  by  the  following 
geometric  property:  Any  tangent  given  by  the  equation  (28)i  is  such  that  if  we 
pass  along  the  curve  in  the  positive  direction  from  the  first  point  of  contact  to 
the  second,  we  do  not  pass  through  any  point  at  which  the  tangent  is  parallel 
to  the  double  tangent;  whilst  for  any  double  tangent  of  the  second  set,  given  by 
a  root  of  (28)a,  we  always  pass  through  one,  and  only  one,  point  at  which  the 


14  ON  THE  PENTACARDIOID. 

tangent  is  parallel  to  the  double  tangent.  Since  each  of  these  equations  is  self- 
conjugate  and  of  odd  degree,  it  follows  that  every  C*  has  at  least  two  double 
tangents  and,  when  there  are  only  two,  they  belong  to  two  different  sets. 

It  should  be  noted  that  if  we  consider  Sj  as  the  variable  and  i  as  a  parameter, 
then  (28)„  is  the  equation  of  a  family  of  lines,  or  rather  of  two  families  of  lines, 
and  the  envelopes  of  these  lines  are  precisely  the  two  deltoids  (25)„  already 
found  in  connection  with  the  tacnodes.  That  this  is  the  case  follows  from  the 
geometric  connection  between  the  double  tangents  of  a  curve  and  the  tacnode; 
analytically  it  is  shown  very  simply  by  the  following  form  of  the  discriminant 
of  (28)„,  which  as  usual  gives  the  values  which  the  variable  (the  variable  here 
being  Sj)  takes  for  all  points  on  the  envelope  of  the  system.  Calling  t^,  t^,  t^ 
the  roots  of  (28),,,  we  have 

<i  +  <2  +  '3  =  (2/3K(e*"  +  l);         Ws  =  ^  "•  (29) 

Now  if  two  roots  become  equal,  we  can  put  ti=.L,  and  therefore  t^^=s"lt\; 
so  that  the  map  equation  of  the  discriminant  is 

2<i  +  eV<f  =  (2/3)si(f^"  +  l),  •     (30)„ 

which  being  solved  for  s^  gives 

s,  =  (3/2)(2<,  +  e"/<f)/(f*"  +  1) ;  (31)„ 

and  if  we  substitute  e"<  for  <i,  we  have  an  equation  identical  with  (26),„  and  this 
proves  the  fact  in  question. 

For  certain  values  of  Sj,  (28),,  becomes  a  perfect  cube  and  three  double 
tangents  unite  and  have  contact  at  one  point  only,  which  is,  however,  a  double 
point  of  the  curve ;  when  this  happens,  we  have  a  C^  with  an  oscnode.  From 
equations  (29)  we  see  that  (28)„  is  a  cube  only  when  if  =  c",  which  requires  that 
»i  should  have  one  of  the  two  values  (9/2)e"/7(«*"  +  1)>  ^=  1>  2. 

Note  (added  June,  1908).  For  completeness  we  should  add  the  following 
compound  singularities : 

Three  coincident  double  tangents  giving  a  triple  tangent;  the  necessary  and 
sufficient  condition  is 

sj  =  —  (3/2)[(l  +  e")<  +  l/<2](e2"  +  f*")  ;  n  =  1,  2,  3  or  4.  (32) 

Four  coincident  double  tangents;  this  singularity  includes  a  tacnode  and 
may  be  called  a  tacnodal  triple  tangent.     The  condition  is 

s,  =  —  (3/2)(e"  +  £*"),  or  Sj  =  —  (3/2)(2  +  e")(e^"  +  e*") ;    w  =  1,  2,  3  or  4.    (33) 


::?•- 


C4) 


^; 


U) 


(e-l 


If} 


Ul 


U) 


(^) 


{^J 


Fio.  III.  — Compound  singularities  of  tlie  C. 

(15) 


16  ON  THE  PENTACARDIOID. 

Six  coincident  double  tangents  giving  a  quadruple  tangent.  The  condition  is 

«i  =  —  3/2.  (34) 

For  convenience  I  append  a  table  of  all  the  compound  singularities,  including 
those  mentioned  in  the  above  note,  and  give  references  to  the  figures  in  which 
occur  the  several  singularities  and  the  penultimate  forms  of  the  same.  I  should 
remark  that  the  curves  in  Figure  III  are  schematic,  but  resemble  quite  closely 
the  actual  curves. 

Singularities.  Condition  on  Si-  Figures. 

1.  Triple  point,  1  branch  *i=2<— l/i^  HI,  (<^) 

=  2  cusps,  1  node 

2.  Triple  point,  2  branches         Sj  =  .  ?  .  II,  (i);  III,  (e) 

=  2  nodes,  1  cusp 

3.  Quadruple  point,  1  branch     Sj  =  —  3  III,  [i)  and  (/) 

=  3  nodes,  3  cusps 

4.  Tacnode  Si  =  — (3/2)(2< 

=  2  nodes,  2  double  tgs.  +  f7<2)(e''"  +  6*")     I,  (d) 

5.  Oscnode  «!  =  —  (9/2)(e"  +  e*")  HI,  (n)  and  (o) 

=  3  nodes,  3  double  tgs. 

6.  Triple  tangent  Si  =  —  (3/2)[(l  +  £")< 

=  3  double  tangents  +  1  /t^lie^"  +  f*")     III,  (a)  and  (/) 

7.  Tacnodal  triple  tangent         s,  =  —  (3/2)(e'"  +  f'")  HI,  (b),  (j)  and  (0 

=  2  nodes,  4  double  tgs.     or  —  (3/2)(2  +  e'')(e^"+e*») 

8.  Quadruple  tangent  Si  =  — 3/2  HI,  (»«) 

=  6  double  tangents 

.    §  4.     The  Complete  System. 

Thus  far  I  have  considered  the  properties  of  the  G^  in  a  somewhat  isolated 
manner;  it  is  my  purpose  now  to  take  up  the  sets  of  C^'s  which  are  connected 
together  by  the  possession  of  certain  special  properties  or  certain  common  forms 
of  singularities,  and  I  shall  devote  this  section  to  what  may  be  called  the  com- 
plete system  of  C^'s.  By  the  complete  system  I  mean  any  collection  of  curves 
including  within  itself  all  the  double  infinity  of  forms  or  shapes  which  the  C^ 
can  have.     In  connection  with  this  system  of  C^'a  I  shall  consider  certain  singly 


ON  THE  PENTACARDIOID.  17 

infinite  sub-systems,  and  also  certain  interesting  regions,  loci  and  envelopes 
which  naturally  present  themselves  in  the  course  of  the  investigation  and  some 
of  which  we  have  already  come  across  in  deriving  the  equations  of  condition 
treated  in  the  previous  section. 

The  simplest  analytical  expression  for  this  complete  system  is  the  standard 
equation  which  we  have  been  using  above,  namely 

Sx  =  —  12t—6s2t^  +  4Sit^—3t\  (15) 

or,  in  line  form, 

Sx  +  15t  +  I0s2<2—  10s/  +  15<*  +  3t/t^  —  0.  (20) 

When  in  either  of  these  equations  we  allow  the  determining  coefficient  s^,  or, 
more  conveniently  for  our  purpose,  Sj,  to  take  the  double  infinity  of  values  of 
the  complex  number  of  the  binary  domain,  we  get  all  possible  types  of  (7^  We 
mp,y  therefore  consider  Sj  as  representing  a  point  of  the  plane,  and  two  things 
are  to  be  noted :  first,  that  though  s,  is  subject  to  the  condition  that  it  must 
always  be  the  sum  of  three  turns  or  of  one  turn  and  a  pair  of  inverse  points, 
this  condition  does  not  impose  any  restriction  on  the  value  of  s-i ;  and  secondly, 
that  owing  to  the  relation  between  the  determining  coefEcient  and  the  shape  of 
the  curve  established  at  the  end  of  §  1,  we  may  without  any  loss  of  generality 
restrict  Sj  to  any  third  portion  of  the  plane  bounded  by  a  pair  of  straight  lines 
or  rays  through  the  origin.  We  shall  make  use  of  this  restriction  on  Sj  later  on. 
I  shall  begin  this  investigation  with  the  cusps.  The  first  fact  that  strikes 
our  attention  is  that  though  we  have  a  double  infinity  of  curves  yet  they  have 
but  a  single  infinity  of  cusp  tangents ;  for  on  combining  the  condition  for  cusps, 
which  is  simply 

l-\-8zt  —  Sit^  +  i^=0,  (35) 

with  the  line  equation  (20)  of  the  system,  both  Sj  and  s^  are  eliminated  together, 
giving  as  the  envelope  of  the  cusp  tangents  the  curve 

Sx+ 5t+ 5t^  +  Syt^=0..  (36) 

This  is  a  three-cusped  epicycloid  concentric  with  the  whole  system  (20)  and  has 

for  its  map  equation 

Sx  =  —  4t  — 1\  (37) 

Each  line  of  this  curve,  therefore,  must  be  a  cusp  tangent  for  a  whole  infinity 
of  C^'s  of  the  system.  But  it  must  be  noted  that  not  any  three  tangents  of  (36) 
can  be  chosen  as  cusp  tangents  of  a  C^  of  the  system ;  for  the  cusp  parameters 


18  ON  THE  PENTACARDIOID. 

of  any  curve  of  the  system  are  subject  to  the  involution  determined  by  equation 
(35),  namely  Sg  -]-  1  =  0,  and  therefore  when  two  of  the  tangents  are  chosen,  the 
third  is  uniquely  determined. 

The  epicycloid  (36)  has  some  further  interesting  connections  with  the  cusps, 
which  I  shall  here  set  forth.  For  this  purpose  I  shall  consider  the  behavior  of 
the  three  cusps  when  we  hold  one  of  the  cusp  parameters  fixed.  The  cusps  of 
the  complete  system  are  all  included  in  the  formula 

3c„  =  -  9<„  —  Ss^tl  +  Sitl,  n  =  l,2,  3,       (38) 

where  <„  is  a  root  of  (35).     To  study  the  behavior  of  the  cusps,  it  is  best  to 

replace  Sj  and  s^  by  their  values  in  terms  of  <„  and  to  simplify  the  equations  by 

means  of  the  involution  Sg  +  1  =  0.     We  thus  obtain,  as  the  equations  of  the 

three  cusps, 

3ci  =  —  6^1  +  2tl/t^  —  2t%  +  t{,  ' 

Sc^  =  —  6<2  +  2tl/t,  —  2tlt,  +  4,    ■  (39) 

3c3  =  —  6^3  +  2tyti  —  2tlt,  +  tl  . 

This  of  course  is  not  the  only  form  in  which  the  equations  can  be  put ;  and  by 
replacing  one  of  the  parameters  by  its  value  derived  from  the  involution 
Sg  +  1  =  0,  we  get  another  form  which  will  be  found  useful  to  us,  namely 

3c3  =  6/t,t,  +  2/tltl  +  2lt\tl  +  \lt\tl,  (40) 

and  two  similar  equations  obtained  by  cyclically  interchanging  the  subscripts. 

Now  let  us  hold  one  of  the  parameters,  t-^  say,  fixed  and  allow  the  other  two 
to  vary,  replacing  them  by  t.  Under  these  conditions  the  three  cusps  will  have 
to  move  along  definite  curves,  since  their  positions  depend  on  the  single  variable  t. 
The  cusp  Ci  is  by  this  means  separated,  as  it  were,  from  the  other  two,  and  these 
latter  lose  their  identity  and  may  be  treated  as  being  a  pair  interchangeable  at 
will.     The  cusp  Cj  moves  along  one  curve,  namely 

3ci  =  —  6<i  +  2t\jt—  2t\t  -\-t\,  (41) 

and  the  other  two  move  along  a  second  curve  given  by 

3c  =  —  6<  +  2tyti  —  2t\  +  <*  (42) 

or  the  equivalent  equation 

3c  =  6/<i^  +  2lt\e  +"  2lt\t^  +  1  lt\tK  (43) 

Equation  (41),  the  locus  of  Cj,  represents  a  straight  line,  or  rather  a  seg- 
ment of  a  line  described  twice  over;  the  cusps  or  extremities  of  this  segment 


ON  THE  PENTACARDIOID.  19 

are  given  by  the  values  of  t  determined  by  the  equation  dci/dt=  0;  i.  e.,  by 
t%  +1  =  0.     The  ends  of  the  segment  are  therefore 

3^  =  —  6<i  ±  4i<P  +  t\.  (44) 

The  length  of  this  segment,  or  the  distance  between  the  two  points  given  by 
(44),  is  8 ;  and  it  should  be  noted  that  this  is  independent  of  the  value  of  t^. 

It  is  quite  clear,  apart  entirely  from  the  equation  (41),  that  the  cusp  Cj  must 
move  along  a  straight  line  when  t^  is  fixed,  since  this  means  that  the  cusp  tangent 
on  which  Cj  lies  is  a  definite  fixed  tangent  of  the  epicycloid  (36).  The  above 
results  give  us  this  further  information,  that  the  cusp  does  not  travel  along  the 
whole  line  but  is  restricted  to  a  finite  portion  of  length  8,  no  matter  what 
tangent  of  (36)  we  may  choose  as  the  cusp  tangent. 

The  other  two  cusps  move  along  the  curve  (42)  or  (43).  This  curve  is  an 
octavic  with  two  cusps  given  by  the  same  values  of  t  which  determine  the  cusps 
of  the  segment  (41);  namely,  by  t=.  dzijtl'^;  the  cusps  of  the  curve  (42)  are 
therefore 

ZG=±4iltY^-l/tl.  (45) 

Now  suppose  we  give  another  value  to  ti;  we  shall  evidently  get  two  new  curves, 
another  segment  for  Cj  and  another  octavic  for  Cj  and  Cj ;  and  if  we  suppose  t^  to 
vary  continuously,  we  shall  get  two  families  of  curves,  and  the  cusps  of  the  two 
families  will  lie  on  certain  curves;  namely,  the  curves  obtained  by  letting  t^  be  a 
variable  in  equations  (44)  and  (45);  but  (45)  is  merely  the  equation  of  the  epi- 
cycloid (36)  in  slightly  different  form,  as  is  clear  when  we  make  the  substitution 
t  =  i/ty^;  hence  this  epicycloid  is  not  only  the  envelope  of  the  segments  (41)  or 
cusp  tangents,  but  also  the  locus  of  cusps  of  the  octavics  (42).  In  order  to  see 
what  relation  this  epicycloid  has  to  the  singularities  of  the  system  of  C^'s,  we 
must  look  a  little  more  closely  at  the  values  of  t  which  give  the  cusps  of  (42). 
As  we  saw  above,  these  values  are  the  roots  of  the  equation  t^f  +  1  z=.  0,  where 
t  stands  for  t^  or  tg ;  but  tjt^ts  +1  =  0  always,  and  therefore,  whenever  Cg  or  Cg  is 
at  a  cusp  of  (42),  tz  =  t^  and  the  two  cusps  coincide,  so  that  we  have  a  triple 
point  of  the  kind  described  on  page  11.  Thus  we  see  that  the  epicycloid  (36)  is 
the  locus  of  all  such  triple  points  occurring  in  the  complete  system  of  C"s.  This 
fact  can  be  proved  more  directly  by  putting  t  =  tj^  in  equation  (42),  for  then  Cj 
coincides  with  either  c^  or  Cg,  and  at  the  same  time  the  equation  becomes  iden- 
tical with  (36).  From  equation  (40)  we  see  that  when  q  and  Cg  coincide,  Cg  is  on 
the  curve 

3x  =  6<«  +  4<»  +  fi,  (46) 


20  ON  THE  PENTACARDIOID. 

obtained  by  putting  ^j  =:  ^g  =  Ijt  in  (40) ;  and  this  is  the  same  curve  as  (44),  since 
the  two  equations  become  identical  when  we  make  the  substitution  f  ^  —  <i. 

It  is  an  interesting  fact  that  the  complete  envelope  of  the  family  of  octavics 
(42)  consists  of  both  the  curves  (37)  and  (46).  The  envelope  can  be  obtained  by 
the  ordinary  process ;  but  this  is  much  simplified  by  noting  that  equation  (43)  is 
symmetric  in  t  and  ti,  and  when  this  is  the  case  the  envelope  is  given,  at  least 
partially,  by  merely  putting  <=  <i;  This  gives  the  curve  (46).  In  order  to  get 
the  other  part  of  the  envelope,  we  must  employ  the  usual  process  and  apply  it 
to  equation  (42),  when  we  obtain  equation  (36).  It  should  be  noted  that  if  we 
use  only  one  form  of  the  equation,  we  get  only  a  part  of  the  envelope,  and  to 
get  the  complete  envelope  we  must  use  both  (42)  and  (43);  this  is  due  to  the 
fact  that  the  epicycloid  (36)  is  not  only  a  part  of  the  envelope  but  also  the  cusp 
locus,  and  unless  special  precautions  are  taken  the  values  of  the  parameter 
giving  the  cusps  factor  bodily  out  of  the  equation  of  condition  for  the  envelope. 
The  curve  (46),  besides  being  the  partial  envelope  of  the  octavics,  is  also  the  locus 
of  the  ends  of  the  segments  along  which  the  cusps  of  the  G^'&  travel,  and 
therefore  bounds  the  region  of  the  plane  within  which  all  the  cusps  of  the  com- 
plete system  lie.  We  see,  as  a  corollary  of  the  theorem  of  §  1,  that  the  curve 
(46)  must  have  triple  symmetry ;  this  is  also  proved  independently  by  merely 
substituting  i^t  for  t  in  (46)  and  noticing  that  we  then  get  Sax  in  place  of  3a;. 

The  rest  of  this  section  will  be  devoted  to  an  investigation  of  the  reality  of 
the  singularities  of  the  C^,  and  I  shall  set  forth  somewhat  fully  a  graphical 
method  for  the  determination  of  this  point  arising  from  the  equations  of  condition 
expressed  in  §  3.  The  problem  then  is  this:  Given  the  equation  of  a  G^,  how 
many  real  singularities  has  it?  I  shall  take  first  the  cusps,  then  the  double 
tangents,  and  lastly  the  nodes.  Let  then  a  definite  curve  be  given,  and  I  shall 
suppose  that  its  equation  has  been  reduced  to  the  standard  form  (15)  or  (20) 
given  above.     The  cusp  condition  is  then  the  simple  expression 

1 +  S3<  — «/  +  !!=' =  0,  (35) 

and  the  reality,  coincidence  and  so  forth  of  the  cusps  depend  on  the  reality, 
equality  and  other  relations  of  the  roots  of  this  equation,  and  these  relations 
depend  entirely  on  the  value  of  the  coefficient  Sj.  As  in  a  previous  case  (p.  14), 
so  here  also  we  may  consider  (35)  as  the  equation  of  a  family  of  straight  lines, 
Sj  being  the  variable  and  t  the  parameter  of  the  family ;  (35)  is  therefore  the 
line  equation  of  a  curve,  and  the  discriminant  of  (35)  considered  as  a  cubic  in  t 


ON  THE  PENTACAEDIOID.  21 

will  be  the  equation  in  map  form  of  the  point  curve,  or  envelope  of  the  family 

of  lines.     Instead  of  forming  the  discriminant  by  the  ordinary  methods  leading 

to  an  equation  involving  only  Sj  and  s^  as  the  variables,  we  follow  the  far  simpler 

method  of  expressing  the  discriminant  in  terms  of  Sj  and  t;  and  thus  it  is  that 

we  get  the  equation  in  map  form.     If  it  were  desirable,  we  could  eliminate  t 

between   this  resulting  equation   and   its  conjugate    and  obtain    the  ordinary 

equation  of  the  curve.     Since  Sj  =  ^j  -f  ^3  -f  ^3  and  t^t^fs  =  — 1,  the  discriminant 

is  simply 

si  =  2t—l/t\  (47) 

and  this  represents  a  three-cusped  hypocycloid  or  deltoid.  Now  the  deltoid  is 
a  curve  which  has  been  carefully  studied  and  whose  properties  are  well  known,* 
and  so  it  may  be  used  with  profit  in  the  present  investigation.  Since  for  a  fixed 
value  of  Sj  the  three  roots  of  (35)  are  the  parameters  of  the  cusps  of  the  corre- 
sponding C^,  and  at  the  same  time  the  parameters  of  the  tangents  from  the 
point  Si  to  the  deltoid  (47),  we  see  at  once  that  the  C^  determined  by  any 
particular  value  of  Sj  has  as  many  real  and  distinct  cusps  as  there  are  real  and 
distinct  tangents  from  the  point  Sj  to  the  deltoid.  Hence  when  s^  is  within  the 
deltoid,  the  C^  has  three  real  and  distinct  cusps;  when  it  is  outside  the  deltoid, 
the  G^  has  only  one  real  cusp ;  and  when  Sj  is  on  the  deltoid,  the  C^  has  two 
coincident  cusps  and  therefore  a  triple  point.  There  are  three  singular  points 
on  the  deltoid,  namely,  the  three  cusps,  for  which  (35)  becomes  a  perfect  cube ; 
and  hence  when  Sj  is  at  one  of  these  points,  the  three  cusps  of  the  C^  must 
coalesce,  and  therefore  the  curve  will  have  a  quadruple  point.  Since  the  three 
cusps  of  the  deltoid  are  equispaced  about  the  center,  which  in  this  case  is  the 
origin  of  coordinates,  the  three  special  values  of  Sj  differ  only  by  a  cube  root  of 
unity,  and  hence  therie  is  only  one  C'^  having  three  coincident  cusps.  Thus  the 
position  of  Sj  in  reference  to  the  deltoid  (47)  tells  graphically,  and  at  a  single 
glance,  the  story  of  the  cusps  of  the  corresponding  C^. 

Next  let  us  examine  the  double  tangents.  We  have  already  seen  (p.  14), 
that  the  discriminant  of  the  double  tangents  consists  of  two  other  deltoids; 
namely, 

s,  =  -  (3/2)(2<  +  e-ie){^^'  +  e'").  (3l)„ 

*  Of.  J.  Steiner:  Ueber  eine  Kurve  Dritter  Klasse,  Crelle's  Journal,  Vol.  LIII  (1857),  pp.  331  ff.,  where  the 
curve  is  studied  by  the  methods  of  synthetic  geometry.  For  a  treatment  of  the  curve  along  the  lines  of  the 
analysis  employed  in  the  present  paper,  cf.  F.  Morley :  Orthocentric  Properties  of  the  Plane  n-Line,  Transactions 
of  the  American  Mathematical  Society,  Vol.  IV  (1903),  pp.  1  ff.  ;  »nd  H.  A.  Converse  :  On  a  System  of  IlypocycloidB 
of  Class  3,  Annals  of  Mathematics ^  Series  8,  Vol.  V  (1904) 


22 


ON  THE  PENTACAEDIOID. 


The  position  of  Sj  in  relation  to  these  two  deltoids  tells  the  story  of  the  double 
tangents  in  the  same  way  as  its  relation  to  (47)  told  the  story  of  the  cusps. 
When  si  is  on  either  of  the  deltoids,  the  C^  has  a  tacnode;  when  s^  is  on  both 
deltoids,  the  C^  will  have  two  tacnodes;  and  when  it  is  at  a  cusp  of  either 
deltoid,  the  C^  will  have  three  coincident  double  tangents  with  a  single  pair  of 
contacts,  and  this  singularity  is  the  oscnode. 

Before  proceeding  to  the  nodes,  it  will  be  convenient  to  examine  the  relative 
size  and  position  of  the  three  deltoids  just  considered.  For  this  purpose  we  will 
compare  the  three  equations  (47),  (26)i  and  (26)2.     I  shall  call  the  discriminant 


Fig.  IV.  — The  discriminant  deltoids  of  the  complete  system  of  C's. 

of  the  cusps  Z?o  and  the  discriminants  of  the  double  tangents  or  nodes  D^  and  D^. 
Since  there  is  no  constant  term  in  the  three  equations,  it  follows  that  the  three 
deltoids  are  concentric;  and  as  the  coefficients  are  all  real,  it  follows  that  the 
axis  of  reals  is  an  axis  of  symmetry  for  each  of  the  deltoids.  For  the  further 
determination  of  the  curves,  we  need  merely  determine  the  position  of  one  of  the 
cusps  of  each  curve.  The  cusps  on  the  axis  of  reals  are  at  the  following  points; 
On  Do  at  —3,  on  D^  at  9cos(27t/5)  =  — 2.78  +,  on  D2  at  9cos(47t/5)  =  7.28  +. 
Thus  the  cusps  of  Dq  and  D^  are  pointed  in  the  same  direction,  whilst  those  of  D^ 
are  pointed  in  the  opposite  direction ;  hence  Dq  and  D^  do  not  intersect  each 
other  at  all,  whilst  Dg  intersects  both  Dq  and  D^  in  six  points  each.  The  three 
curves  are  therefore  as  shown  in  Figure  IV. 


ON  THE  PENTACARDIOID.  23 

These  deltoids  divide  the  plane  into  a  certain  number  of  regions,  and  the 
number  of  real  singularities  possessed  by  any  given  C^  depends  on  the  region 
in  which  the  coefficient  Sj  of  the  given  C^  lies.  We  need  not  consider  all  the 
regions,  since,  owing  to  the  theorem  deduced  at  the  end  of  §  1,  the  shape  and 
therefore  all  the  singularities  of  the  C^  are  the  same  for  any  three  values  of  s, 
which  are  equispaced  about  the  origin;  and  if  two  values  of  Sj  are  reflexions  of 
each  other  in  any  one  of  the  three  axes  of  symmetry  of  the  deltoids,  then  the 
two  corresponding  C^'s  are  images  of  each  other  and  therefore  have  exactly  the 
same  singularities.    Hence,  in  considering  the  nature  of  the  singularities,  we  can 


Fio.  v. —The  sub-regions  of  s,. 

restrict  s^  to  any  portion  of  the  plane  bounded  by  two  rays  from  the  origin 
making  an  angle  of  60  degrees  with  each  other.  For  convenience,  we  choose 
for  the  limiting  rays  the  positive  half  of  the  axis  of  reals  and  the  ray  through 
the  point  — w^,  both  limits  being  included  in  the  region  of  Sj .  This  region  may 
be  called  the  essential  region  of  s^.  And  now,  by  means  of  the  three  deltoids, 
this  region  is  divided   into  six  sub-regions  which   will  be   designated   by  the 

symbols  R^, ,  R^.     These  regions  are  shown  in  Figure  V,  in  which  the 

relative  dimensions  of  the  three  deltoids  have  been  slightly  altered  for  the  sake 
of  clearness.  R^  is  infinite  in  extent  and  includes  all  that  portion  of  the  essential 
region  of  Sj  which  lies  entirely  outside  of  all  the  deltoids. 

By  invoking  the  principle  of  continuity  and  arguing  from  what  we  may 
know  about  the  C*  for  a  particular  value  of  «i,  we  can  by  means  of  these  regions 
determine  almost  everything  about  the  singularities;  the  only  difficulty  arises  in 


24  ON  THE  PENTACARDIOID. 

trying  to  determine  the  actual  number  of  nodes.  By  enumerating  the  number 
of  tangents  that  can  be  drawn  to  the  deltoids  from  a  point  in  any  one  of  the  six 
regions,  we  know  at  once  how  many  cusps  and  double  tangents  belong  to  the 
corresponding  C".  With  regard  to  the  nodes,  we  can  not  say  how  many  there 
are  for  each  position  of  S],  but  we  can  tell  whether  the  G^  has  an  odd  or  an 
even  number.  For  the  only  way  in  which  the  C^  can  acquire  or  lose  nodes  is 
by  passing  through  some  intermediate  complex  singularity  involving  one  or  more 
nodes;  these  singularities  are  the  following:* 

1st,  Triple  point  with  a  single  branch  ;  2nd,  Quadruple  point  with  one 
branch  ;  3rd,  Tacnode  ;  4th,  Oscnode;  and  5th,  Triple  point  with  two  branches. 
In  the  first  case  the  C^  loses  or  gains  one  node  ;  in  the  second,  three  nodes;  and 
in  the  last  three  cases,  two  nodes.  Hence  we  may  state  generally  that  when 
the  C^  acquires  a  node  for  values  of  Sj  other  than  those  given  by  Z>o,  it  acquires 
a  pair  of  nodes.  Now  when  Sj  ^  0,  we  have  the  three-cusped  epicycloid,  which 
we  know  independently  has  no  nodes  ;  therefore  when  «i  is  within  Dq  the  C^  has 
an  even  number  of  nodes,  and  when  Sj  is  without  Dq  the  G^  has  an  odd  number 
of  nodes.  We  can  now  make  out  a  table  of  singularities  for  the  various  regions, 
the  number  of  simple  singularities  possessed  by  a  given  C^  being  placed  opposite 
the  region  in  which  the  coefiGcient  Sj  of  the  given  C^  happens  to  lie. 


Region. 

Cusps. 

Db.  Tgs. 

Nodes 

R,    ■ 

3 

6 

Even 

R, 

3 

4 

Even 

Rs 

3 

4 

Even 

R, 

3 

2 

Even 

A 

1 

4 

Odd 

i?« 

1 

2 

Odd 

This  table  of  regions  is  incomplete  in  so  far  as  the  number  of  nodes  is  con- 
cerned and  must  remain  so  until  the  equation  of  condition  for  the  singularity 
consisting  of  two  nodes  and  a  cusp  is  found.  I  shall  here  leave  the  subject, 
with  the  hope  that  it  may  be  completed  at  a  later  date. 

Johns  Hopkins  Univkbsitt.  March,  1908. 

•  Cf.  Table  of  singularities,  p.  16.  The  completeness  of  this  table,  upon  which  rests  the  validity  of  the 
argument  here  used,  has  not  as  yet  been  rigidly  proved;  but  careful  investigation  makes  it  almost  certain  that 
there  are  no  other  compound  singularities  occurring  on  any  &'. 


BIOGRAPHICAL  NOTE. 

Edward  C.  Phillips  was  born  in  Germantown.  Pennsylvania,  on  November 
4,  1877.  His  early  education  was  secured  in  the  Parochial  Schools,  and  he  made 
his  collegiate  studies  at  the  College  of  St.  Francis  Xavier,  New  York  City, 
graduating  from  that  institution  with  the  degree  of  Bachelor  of  Arts  in  1898. 
He  then  entered  the  Novitiate  of  the  Society  of  Jesus  at  Frederick,  Maryland. 
PVom  1901  to  1904  he  was  at  Woodstock  College,  Maryland,  engaged  chiefly  in 
graduate  studies  in  Philosophy.  In  October,  1904,  he  came  to  the  Johns  Hop- 
kins University  and  entered  the  department  of  Mathematics  as  a  graduate 
student.  Since  then,  with  the  exception  of  the  year  1906-7,  he  has  been 
following  courses  of  Mathematics,  Physics  and  Physical  Chemistry. 

March,  1908. 


BERKELEY  ^'^^^' 

^HIS  BOOK  IS  due"^  ^^_      ^^ 
ao  «-^  no.  rl^^=^  BeS^^^«^  bate 


20m-ll,'20 


I- 


^. 


